p-stable group

In finite group theory, a p-stable group for an odd prime p is a finite group satisfying a technical condition introduced by Gorenstein and Walter (1964, p.169, 1965) in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups.

There are several equivalent definitions of a p-stable group.

We give definition of a p-stable group in two parts.

The definition used here comes from (Glauberman 1968, p. 1104).

Let p be an odd prime and G be a finite group with a nontrivial p-core

(

Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that

is a normal subgroup of G. Suppose that

x ∈

is the coset of

Now, define

as the set of all p-subgroups of G maximal with respect to the property that

Let G be a finite group and p an odd prime.

Then G is called p-stable if every element of

is p-stable by definition 1.

Let p be an odd prime and H a finite group.

Then H is p-stable if

and, whenever P is a normal p-subgroup of H and

If p is an odd prime and G is a finite group such that SL2(p) is not involved in G, then G is p-stable.

If furthermore G contains a normal p-subgroup P such that

is a characteristic subgroup of G, where

is the subgroup introduced by John Thompson in (Thompson 1969, pp.